Out of about 100 rounds today (this was over two hours), I got about 15 'double-shots'. The first double-shots even occured during the first clip. I thought at first I'd put too much lub in, but after wiping it down two/three times..

I tried pulling more 'decisively' on the trigger to preclude possible bounce, changed my position on the pistol grip...

Bushmaster. I haven't messed with it ... so I assume it is factory. The guy I got it from said he'd only test shot it after he assembled it.
Concerning the trigger catching the hammer... when I've got it open and pull the hammer back, it definitely catches. I'm going to take a look at the parts list to identify the "disconnector". Is the alignment adjustable, or is it just replacable.

It's not the ammo.
You probably have some out of spec parts. I had the same thing happening to me once. My hammer and another part were out of spec. S.A.W. replaced them and that stoped the prob.
My second round would fire when the trigger was "let off" after the first shot. In other words, I could take the first shot and hold the trigger all the way back. When I released the trigger, the hammer would fall on the round that had been chambered.
Have a gunsmith check it.

I'm having the exact same problem. My gun will fire a second time when I let off the trigger. I've swapped three complete sets of lower receiver parts through the gun, and the problem persists. The first set was factory Colt, in my Colt gun. The second set was also Colt parts. I tried a set of Bushmaster parts, with the same problem. The only two parts I have not swapped each time is the trigger spring and the disconnector spring? Could it possibly be one of these parts or both causing the trouble?

SOMETIMES you have to grind a little of the front of the disconnector where it rests on the trigger. You see? The spring pushes it up, and it pivots on the trigger pin and rests on the trigger. There is a little part that keeps it from going too far forward. If you remove a little metal there, the disconnector can more engage the hammer a little more.
Your receiver could have the hammer and/or trigger holes slightly off causing this. I have seen this before, resulting in the problem you are having.
OR, the front of the trigger, or the hammer sear surface could be excessively worn.
But if you replaced the hammer AND trigger AND disconnector several times, this is not a factor.
With the upper half removed, cock the hammer by hand. Pull the trigger while catching the hammer in your fingers. Now, while holding the trigger, push the hammer back. It should catch on the disconnector.
Now, let go of the hammer. Slowly release the trigger, and the disconnector will let go, and the hammer will be caught by the trigger. If not, there is a problem! The disconnector needs to stay engaged with the hammer a little longer on the release.
So, work these pieces by hand, slowly, watching how it works, or is supposed to work. Compare how they work to a properly working AR15.

A_FREE_MAN knows his stuff.
All my AR's have been timed so that the disconnector releases the hammer just as the trigger is returning fully forward.
In this way all other aspects of the trigger pull can be tweaked until I am satisfied.
And I'm a picky bastard....one ancient gunsmith hereabouts calls me a 'goddam rifle crank' because I suggest I tune his triggers a second time, so that they work properly.
Regards and good shooting!

SF,
Timing is a complex issue and messing with it causes no end of problems for newbs.
Short and sweet, when you thumb cock the hammer, with the trigger forward, the middle hammer hook should ALMOST (it should look like it's going to touch) touch the tip of the disco hook. With the trigger to the rear the hammer should latch on the disco hook and when the trigger is released the hammer should drop from the disco to the sear face of the trigger. Always keep something (I use a finger) between the hammer and the lower so you can't damage something costly if the hammer drops unexpectedly.
As I said tho, it ain't that simple. A simple adjustment, that can sometimes conceal the true problem or even lead to other problems, is to remove enough material from underneath the nose of the disco (where the disco sits on top of the hammer) so that the disco can rock forward enough to properly engage the middle hammer hook in the above check.
If sending your rifle back to the maker for warranty work is an option for you then now is the time to take advantage of it.

Well folks, I got the lower on the way back to Bushmaster last week... I'll let you know how it turns out. Oh yes, thanks to all for the input, it got me to looking at the mechanism... and I appreciate your assistance.